You could certainly do it in brich ply, but...
If you live near a city, find a custom cabinet shop,. Tell them you are an amateur woodworker (so they know you are not their competition) and ask them if they know where you can get 4x8 or 4x10 sheets of A1 veneer core plywood.
Veneer core does not have voids. The glue and skins used strong and light... So you get strength, but end up with a cabinet that is lighter weight, which is handy when you go to hang the cabinets. (even lighter weight than normal plywood)
Birch ply, is dense glue and fine dense skins which end up being very heavy for the mass. (Even heavier than normal plywood).
You get Veneer core ply with finish quality veneers on both sides. Very little sanding needed.
Cost varies with the types of veneers it's skinned with, but most are at what you are thinking about from the box stores for one side finished birch ply. I think I get 4'x10' sheets of 3/4" Maple veneer core for about that price.
Of course Veneered MDF Core and Melamine are a fraction on that (about half the cost)... But with mdf core, it is dense, so they end up heavy.
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Last edited by MAFoElffen; 07-31-2014 at 10:52 PM.